on \(\mathbb {R}^s\times \mathbb {R}^n\times \mathbb {R}\) ; see [1]} for definitions. Moreover, \(\operatorname{epi}f = \lbrace (u,x,\gamma )\in \mathbb {R}^{s+n+1}~|~f(u,x) \le \gamma \rbrace \)  with a parallel definition for \(\operatorname{epi}\tilde{f}^\nu \) . Since
\((\tilde{u}, \tilde{x}) \in \delta \mbox{-}\hspace{-1.70709pt}\mathop {\rm argmin}\nolimits _{u,x} f(u,x) ~~\Longleftrightarrow ~~ \tilde{u} = 0, ~\tilde{x} \in \delta \mbox{-}\hspace{-1.70709pt}\mathop {\rm argmin}\nolimits _{x} f_0(x) + \mathop {\sum }\nolimits _{i=1}^s p_i f_i(x)\) 
